A radiation model for power degradation estimation in earth centered satellites

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OLITECNICO DI

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ILANO

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ASTER

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A Radiation Model for Power

Degradation Estimation in Earth

Centered Satellites

Author: Alberto MAÑERO CONTRERAS Supervisor: Dr. Francesco TOPPUTO Co-Supervisor Simone CECCHERINI

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CHOOL OF

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NDUSTRIAL AND

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NFORMATION

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CIENCES AND

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ECHNOLOGIES

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POLITECNICO DI MILANO

Abstract

Department of Sciences and Aerospace Technologies School of Industrial and Information Engineering

A Radiation Model for Power Degradation Estimation in Earth Centered Satellites

by Alberto MAÑEROCONTRERAS

E

LECTRIC propulsion systems meant an important improvement in the

space sector ever since it was invented. Lately, it has been tested in different applications and it holds capital advantages over the conventional propulsion systems. On the other hand, low-thrust transfers performed with electric propulsion require large times of flight. A major drawback comes along with this fact: high exposure to trapped energized particles in Van Allen Belts. This radiation environment becomes a considerable issue for electronic systems and satellite onboard equipment, as well as a relevant so-lar array degradation, which implies a significant loss of power capability during the mission. The current thesis focuses on the implementation of a fast and reliable radiation model that is meant to be incorporated into an optimization framework in order to minimize the above-mentioned deterio-ration of the photovoltaic cells along an Earth-centered trajectory. The devel-oped model has been validated with the AE9/AP9/SPM tool and SPENVIS, and the basis for the aforementioned optimal control has been stated.

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POLITECNICO DI MILANO

Sommario

Department of Sciences and Aerospace Technologies School of Industrial and Information Engineering

A Radiation Model for Power Degradation Estimation in Earth Centered Satellites

by Alberto MAÑEROCONTRERAS

I

sistemi di propulsione elettrica hanno significato un importante migliora-mento nel settore spaziale. Recentemente, la propulsione elettrica è stata utilizzata per diverse applicazioni ed offre sostanziali vantaggi rispetto ai convenzionali sistemi di propulsione. D’altra parte, i trasferimenti a bassa spinta, tipici degli attuali sistemi propulsivi elettrici, richiedono alti tempi di volo. Di conseguenza, un inconveniente rilevante è dovuto all’esposizione del satellite alle particelle ad alta energia all’interno delle fasce di Van Allen. Il passaggio all’interno di questo ambiente diventa un considerevole prob-lema per quanto riguarda i sistemi e le apparecchiature all’interno del satel-lite; inoltre, può portare ad una rilevante degradazione dei pannelli solari che a sua volta implica una significativa perdita di generazione di potenza durante la missione. La presente tesi si concentra sullo sviluppo di un mod-ello veloce e rappresentativo per quantificare l’esposizione alle suddette ra-diazioni, in modo tale da poter essere utilizzato in un processo di minimiz-zazione del deterioramento delle celle solari lungo una traiettoria in un sis-tema centrato nella Terra. Il modello prodotto è stato validato attraverso il software AE9/AP9/SPM e con SPENVIS; al contempo, le basi per sviluppare il suddetto controllo ottimo sono state derivate.

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Acknowledgements

First of all, I would like to express deepest gratitude to my advisor Prof. Dr. Francesco Topputo for his support, guidance, understanding and encourage-ment throughout my study and research. Secondly, I owe special recognition to Simone Ceccherini. Without his incredible patience and timely wisdom and counsel, my thesis work would have been a frustrating and overwhelm-ing pursuit.

Now that my student episode approaches to its end, I would like to dedicate some lines to the core of my success: my family. Papá, mamá y hermanita. Ya sabéis que no hay palabras de agradecimiento que describan la gratitud que siento por vuestro apoyo incondicional durante estos ocho largos años. Porque gracias a vosotros, entendí que no importa el número de veces que caigas, mientras siempre te levantes una más. Porque cuando no creí en mi mismo, vosotros fuisteis los que me disteis una inyección de fe. Estoy tremendamente orgulloso de la educación que me habéis proporcionado y de las oportunidades de las que he podido disfrutar gracias a ello. Gracias de corazón.

Furthermore, I would like to acknowledge many people from Milan, but es-pecially, those ones closest to me during these two years. Mohit, Aroon, and Semih. Thank you guys for being a constant in the happiest moments, but most importantly, the saddest ones. It would have been very tough without you all.

I owe special gratitude to my flatmates. They have made me feel comfortable at home and they are the ones who heard my screams and the manic laugh-ters while working on this thesis. Andrés, Yasmina, and especially María, whom I have shared with these two years. Thank you all for your support. En un párrafo aparte, hago mención a los bicchieri. Habéis aparecido tarde, pero habéis sido una bendición. Os podría agradecer muchas cosas, pero en especial, gracias por aguantar mis turras sempiternas sobre la tesis. Habéis sido un pilar fundamental estos meses y espero que esta amistad se pro-longue en el tiempo con más planes en la Península Ibérica.

Para acabar, como no podía ser de otra manera, me acuerdo de vosotros: los buhítos. Porque aunque estemos lejos, crezcamos y cada uno de nosotros tomemos caminos diferentes, nunca dejaréis de ser el motor de mi vida. Gra-cias a todos y cada uno de vosotros, de corazón, por haberme enseñado lo que significa la amistad.

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List of Figures

1.1 Typical orbital transfer from LEO to GEO. . . 6

2.1 Solar flare. . . 13

2.2 Simplified representation of Van Allen Belts. . . 14

2.3 South Atlantic Anomaly. . . 15

2.4 Motion of charged particles captured in the geomagnetic trap. 16 2.5 Coordinate frames. . . 17

2.6 Current-Voltage curve. . . 21

3.1 Proton averaged volume flux comparison.. . . 25

3.2 Proton RDC function. . . 25

3.3 Design strategy flowchart. . . 27

3.4 Linear differential flux distribution.. . . 28

3.5 Logarithmic differential flux distribution. . . 29

3.6 2-D Model of Van Allen Belts. . . 30

4.1 Transfer baseline example. . . 43

4.2 Differential fluence comparison.. . . 44

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List of Tables

2.1 Breakdown Coordinate Transformations. . . 18

2.2 Spectrolab XTJ Conversion Parameters. . . 22

3.1 Recommended Sample Times. . . 28

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Notation

Rn _{n-dimensional Euclidean space} Rn×m _{Matrices of dimension n}_×_m | · | Magnitude of a vector

|A| Determinant of the square matrix A

∇ Gradient of a vector

log Natural logarithm of a vector/matrix

∗ Optimal variable c Exhaust velocity

De,p Relative Damage Coefficient E Energy level

g0 Gravitational acceleration at sea level G Universal gravitational constant Gr Total equivalent flux at 1 MeV hk Altitude value at iteration k H Hamiltonian

Isc Cell short circuit intensity Isp Thruster specific impulse

J Performance index of the optimal problem Jd Julian days

Jr Total equivalent fluence at 1 MeV L Lagrangian

m Mass of the spacecraft ME Mass of the Earth Pmax Cell maximum power RE Radius of the Earth

Sr Switching function for radiation optimal problem u Control variable

Voc Cell open circuit voltage

Fe Matrix of electrons differential flux

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g Gravitational acceleration for the restricted two-body problem In×m Identity matrix n×m

J_Γ Jacobian of the shooting function r Position vector

T Thrust vector v Velocity vector x State vector

y Augmented state-costate vector z Augmented state-costate-STM vector

αp−e Proton-electron conversion parameter α Thrust direction unit vector

Γ Shooting function

δ Altitude step size

Θ Term related with the BC on the states and the terminal cos function

κ Cell coverglass thickness

λ Costates vector

µ Earth gravitational constant

ν Lagrange multipliers for the terminal BC Ξ State Transition Matrix

φ Omnidirectional integral flux

Φ Omnidirectional integral fluence Ψ Terminal function of the states

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Acronyms

COV Calculus Of Variations EFT Equivalent Fluence Theory EP Electric Propulsion

GEI Geocentric Equatorial Inertial GEO Geostationary Earth Orbit GRS Geographic Reference System GTO Geostationary Transfer Orbit

IGRF Inertial Geomagnetic Reference Field

IRENE International Radiation Environment Near Earth IVP Initial Value Problem

LEO Low Earth Orbit

LT2O Low Thrust Trajectory Optimization MAG Geomagnetic

NEO Near Earth Objects

PMP Pontryagin Minimum Principle RAAN Right Ascending of Ascendent Node RDC Relative Damage Coefficient

RK Runge Kutta

SAA South Atlantic Anomaly SEP Solar Electric Propulsion

SPENVIS SPace ENVironment Information System STM StateTransition Matrix

TOF Time Of Flight

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Dedicated to my parents and sister, Alberto, Lola and

Julia, whom I love tenderly.

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1

Chapter

1

Introduction

"One small step for man, one giant leap for mankind."

- Neil Armstrong, Apollo 11

O

CTOBERthe 4th, 1957. History changed from this day forward when the

Soviet Union successfully launched the worldwide known Sputnik I. Some decades after, it is simple to figure how this historical fact impacted a wide range of fields that ushered in new political, military, technological and scientific developments. Most of them were framed during the few follow-ing years from Sputnik’s launch, the Cold War, while space would become another tense arena for this peculiar race, as each side sought to show its great potential and superiority at all possible levels. Barely four months later, the U.S. launched its own artificial satellite, the Explorer I, as a clear oppos-ing argument from the Soviet Union’s aspirations to conquer the space. This satellite is of particular interest for this thesis because it was the first space-craft to detect the Van Allen Radiation Belt, named after its discoverer James Van Allen, returning data until its batteries were exhausted after nearly four months1. In the same year, the President of the United States of America Dwight Eisenhower signed a public order prior to creating the National Aero-nautics and Space Administration (NASA), the well known federal agency dedicated to space exploration. Even though the Soviet Union took another step forward sending Yuri Gagarin as the first man to orbit the Earth in April 1961, the United States effectively ‘won’the space race by landing success-fully on the moon the 20thof July 1969 aboard the Apollo 11, setting the end of a period that left a trace in history.

Almost five decades later, the space sector has experienced an enormous growth. Many other countries have carved their own spot in space as a

1_{https://www.nasa.gov/mission_pages/explorer/explorer-overview.html} _last

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2 Chapter 1. Introduction

way to demonstrate their capabilities of increasing their space activity. Cur-rently, there are thousands of satellites orbiting the Earth while the launch rate is growing. There is a great selection of space applications, ranging from telecommunications to Earth observations. Even though their goals could be diverse, the trajectory analysis and optimization techniques play always a major role.

A trajectory is defined as the path that a moving object follows through space as a function of time. Their applications are mainly placing a satellite into a predefined orbit or either make an orbital transfer from two different orbits. Historically, the study of trajectories has caught the eye of many engineers, especially the study of those that optimize a quantity that normally has a large impact on the overall mission, economically or another kind, such as time, fuel or, as this thesis reports, radiation. Those trajectories are called opti-mal ones.

Another way to distinguish two trajectories resides on the kind of primary propulsion used by the spacecraft. They are broadly divided into two main branches: high-thrust and low-thrust transfers. For the first group, the main propulsive capability of the spacecraft comes from chemical rockets that are modelled as producing instantaneous (ideal) changes of velocity. On the other hand, low-thrust transfers are accomplished through engines that use electrical power to accelerate a propellant and, consequently, to apply a change of velocity to the spacecraft in a very efficient manner; the Electric Propulsion (EP) systems. Even though the idea of electrical propulsion dates back to 1911, it was not developed until 1964 when the experimental satellite SERT-12 _{was launched with an ion engine aboard, operating for 31 minutes. Its} follow-up mission was launched six years later under the name of SERT-23. Nonetheless, high-thrust solutions were more frequently implemented in practical missions at this time due to the not sufficient development of the low-thrust technology. Gradually, EP technology has been improved and re-fined. As a result, its applications have been shifted from station keeping (the first application it had) to primary propulsion solutions. Moreover, it provides many advantages over the conventional propulsion systems. For example, due to the low propellant consumption, electrical propulsion sys-tems aim at a greater payload fraction, cheaper launchers and longer mis-sions. Furthermore, they provide a precise pointing because of their low highly controllable thrust [1]. Unfortunately, the transfers made with this type of propulsion takes very high Times of Flight (TOF), in the order of

2_{https://www.grc.nasa.gov/WWW/ion/past/60s/sert1.htm}_{last visited on 20}th _{of May,}

2018.

3_{https://www.grc.nasa.gov/WWW/ion/past/60s/sert2.htm}_{last visited on 20}th _{of May,}

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1.1. Low-Thrust Optimal Trajectories 3

months depending on the initial and final orbits, and the control authority. Great examples of this technology are the ESA’s SMART-14which was used to test solar electric propulsion and other deep-space technologies, while per-forming scientific observations of the Moon, and the ESA’s Lisa Pathfinder5 (also called SMART-2), which paved the way of future missions by testing in flight the very concept of gravitational wave detection.

Keeping the idea of gathering energy from the Sun as a method to feed the propulsion system, the full development of the Solar Electric Propulsion (SEP) systems becomes particularly important for future space missions, such as NASA Asteroid Robotic Redirect Mission, which will be designed to capture a large (up to 1000 metric ton) boulder from the surface of a Near Earth As-teroid, and bring it into high lunar orbit [2].

1.1 Low-Thrust Optimal Trajectories

During the analysis and design of space missions, some of the most impor-tant tasks are the design and optimization of the transfer path. As stated in [3], trajectory optimization problems that involve high-thrust propulsive systems are typically formulated as discrete optimization problems and gen-erally are well-conditioned from the numerical point of view, however, low-thrust systems operate for large periods of the mission continuously and its associated optimal control problems are numerically ill-conditioned in most of the cases. This thesis mainly focuses on the development of a model of the radiation environment for Earth centered satellites as well as the formulation of the radiation optimal trajectories for low-thrust transfers. Generally, the computation of a spacecraft’s trajectory, particularly the optimal one, is made through the integration of a set of differential equations which represents its dynamics. The terms included in this set of equations normally account for the inertial forces acting on the system as well as other terms linked to control variables such as the thrust, along with its direction, for instance. Hence, the optimal trajectory is the output of an optimization of those control variables. As mentioned before, problems regarding high-thrust propulsion systems can be modeled as discrete ones due to the fact that their propulsive per-formances are bounded in short periods of time, justifying accordingly the fact of the discretization done. Generally, they are straightforward to solve if compared with the low-thrust problems. These latter ones, contrarily, bring complications when it comes to find an optimal solution due to the fact that they do not perform instantaneously. Thus, the optimal solutions of this

4_{http://sci.esa.int/smart-1/}_{last visited on 9}th_{of July, 2018}

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type of problems require tools based on continuous optimization techniques that model the control variables as continuous functions in order to solve the problem appropriately.

Low-thrust optimized trajectories are typically obtained through either in-direct or in-direct methods [42]. Direct methods entail the parametrization of the problemand the use of nonlinear programming techniques to optimize an objective function by adjusting a set of variables. They offer a wide range of advantages because of their robustness and simplicity. Nonetheless, their weak points are both the great amount of time needed to achieve a solution and their poor accuracy which comes along with them. For more information about direct methods, the reader is referred to [4].

Alternatively, indirect methods are based on Calculus of Variations (COV). Classical solutions to minimization problems in the calculus of variations are prescribed by boundary value problems involving certain types of dif-ferential equations, known as the associated Euler–Lagrange equations. The mathematical techniques that have been developed to handle such optimiza-tion problems are fundamental in many areas of mathematics, physics, engi-neering, and other applications. In addition, another principle used to han-dle an optimization problem via indirect methods is the Pontryagin Mini-mum Principle (PMP), which exploits the Euler-Lagrange equations. These two tools are used to transform the original problem to a reduced one, de-pending only on few parameters. This is done by adding constraints rep-resented by Lagrange multipliers λ(t)which are utilized to derive the nec-essary conditions for an extremum considering the variation of the perfor-mance index J. Given that the low-thrust optimal trajectory problems are optimized over time continuously, it is straightforward to understand that the Lagrange multipliers, or costates, are going to be governed by a set of differential equations which will have to be integrated over time. Hence, the conditions for a continuous optimal problem are composed by the ones subjected to the costates λ(t)as well as the ones linked with the state of the system, x(t). Once these conditions are derived, the original problem results in a Two-Point Boundary-Value Problem (TPBVP) that is solved by satisfying the conditions commented above. Indirect methods were named after this fact - solving the canonical problem by transforming it into a TPBVP. Never-theless, these indirect methods are subject to extreme sensitivity to the initial guess of the variables - some of which are not physically intuitive, a fact that comprehends its main disadvantage.

A proper mathematical definition gathering the concept is required once a brief introduction about the low-thrust optimal problem has been given. As stated in [5], this definition reads

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1.2. Motivation of the Thesis 5

Definition 1. An optimal control problem consist on the computation of the state vector x(t) ∈ Rn, the costates vector λ(t) ∈ Rn and the control u(t) ∈_Rm_{based on boundary conditions that optimize some cost function J.} It is of capital importance to dedicate some introduction lines to the cost func-tion J. As explained in [6], when determining the optimal trajectory, this per-formance index J is not always the same. Sometimes, one requirement sets the minimization of the required propellant, giving rise to the minimum fuel problem. In order to solve that problem, it is needed an estimation of the up-per bound of the total transfer time. For that reason, one must solve first the minimum time problem. When one addresses the minimum radiation problem, the same estimation of the final time is a requisite as well. This topic will be covered extensively in Chapter4.

1.2 Motivation of the Thesis

In Section1.1the complexity of solving the optimal low-thrust transfer has been introducted. It does not just comprehend to solve the problem under study subjected to some constraints, but doing it while minimizing a cer-tain cost function to reach the optimal solution. This thesis consists in a follow-up study about these family of optimal trajectories done already for a minimum TOF [6,9], and minimum fuel consumption [7]. As commented previously, the main motivation of this thesis is to compute the trajectory profile that optimizes the radiation absorption, taking into account a model developed with the Space Environment Information System (SPENVIS6) and AE9/AP9/SPM7_{tool, which will be entirely explained in Chapter}₃_.

Therefore, this thesis was conceived to propose a family of solutions for or-bital transfers from diverse LEOs to GEO. It is important to remark that, in opposition to other studies [23], in this thesis, there is going to be only one single transfer orbit which will connect a certain LEO with the GEO. There will not be any hybrid transfer or switching orbit in which both high and low-thrust phases are present, but only the latter one. To clarify this explanation, a scheme of the transfer is depicted in the Figure1.1. Of course, the transfer represented by the dashed line is going to be one of the goals of this thesis, so it is not representative neither realistic the way it is depicted in the picture. This minimum radiation maneuver turns out to be very interesting because under the light of the facts previously explained one would think that min-imum time and minmin-imum radiation problems are inevitably related. As a

6_{The Space Environment Information System. Website:} _{https://www.spenvis.oma.be/}

last visited on 10thof July, 2018.

7_{Radiation Belt and Space Plasma Specification Models. Website:}_{https://www.vdl.afrl.} af.mil/programs/ae9ap9/last visited on 10thof July, 2018.

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matter of fact, it is very straightforward to think that the more the spacecraft holds into the Van Allen Belts, the more radiation it receives. Hence, in a pre-liminary study, both problems must be linked somehow. This topic will be cover in Chapter4while analyzing the statement of the problem.

FIGURE1.1: Typical orbital transfer from LEO to GEO.

Once again, the final aim of this thesis is to compute the optimal trajectory which minimizes the radiation received by the spacecraft. In the next Chap-ter, the radiation environment is deeply investigated, but it is important to comment beforehand in a briefly manner the harsh effects of the radiation on the spacecraft. One variable of capital importance is the Total Ionizing Dose. The term, total ionizing dose, implies the dose is deposited to the elec-tronics through ionization effects only. As stated in [8], the energy deposited by radiation moves the electrons to a higher energy state, thus making them available for conduction and mobile inside a nonconductive material. These electrons, or more correctly the positive charge created by ionization, are the prime cause of the total ionizing dose effects. This ionizing dose not only affect to the electronics of the spacecraft, it also can affect a great variety of equipment types and devices such as optics, electro-optics, photovoltaics, thermal and optical coatings, for instance. It can also damage biological sys-tems, especially astronauts, but that consideration does not play any role in this study because the system is unmanned; a satellite.

In order to solve the problem, a robust solver is needed. The one which has been used during this research consists on an updated version of the Low-Thrust Trajectory Optimization (LT2O) solver developed by Politecnico di Milano [7, 9, 10]. The modifications included in this solver have been the

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1.3. Goals of the Thesis 7

addition of a radiation module for taking into account the fluence of particles along the trajectory of the satellite.

1.3 Goals of the Thesis

The main targets of this thesis are mostly two. In the first place, to develop an accurate and fast radiation model that represents and accounts faithfully for the protons/electrons of a certain energy range present at the Van Allen Belts which the spacecraft would encounter during the transfers under study through variables such as the fluence or the flux. Some verifications must be done in order to double check the model and its efficiency should be ques-tioned as well. Along with that, its simplifications and assumptions must be properly justified, as well as the shortages adjoined to them.

Secondly, once the model has been properly verified and checked, the next potential step is to analyze how the low-thrust optimal control problem can be solved, so implementing a new module into the existent solver of mini-mum time and minimini-mum fuel problems. The solution for different transfer orbits must be studied, typically from several Low Earth Orbits (LEO) to the Geostationary Earth Orbit (GEO). Finally, a trade-off comparing minimum time, minimum fuel and minimum radiation transfer orbits must be done and their differences pointed out.

1.4 State of the Art

Once the main topics of this thesis have been concisely explained, the state of the art of such technology is analyzed in this Section. For example, recent studies indicate new solutions for optimal low-thrust trajectories for SEP [11]. There are mainly three topics which this thesis focuses on, namely radiation environment; how it affects the satellite systems through the model that has been developed, orbital transfers, in particular from LEO to GEO, and the different types of solvers used currently to deal with these kinds of problems. It is well known the huge impact that a harsh environment like the one present in the outer space could have on the most sensitive devices of the spacecraft if they are not well isolated against radiation. From an engineering perspective, these effects are important in determining the reliability of electronics to the ionizing dose environment, for example. Estimations on these effects have been studied for many years in order to be able to preserve their function-ality in the radiation environment found in space [8,12,13]. A lot of effort has been invested as well in the matter of modeling this environment from the numerical point of view, giving rise to several models which have been

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applied to studies to achieve diverse goals. For example, the research in [14] accounts for a minimization of the proton displacement damage dose during electric orbit raising of satellites. A study about radiation optimal transfer from GTO to GEO with EP is presented in [15]. Contrarily, in the work ref-erenced in [16], some radiation damage constraints are imposed to perform a minimum fuel electric orbit raising. Nevertheless, due to the fact that this thesis is mainly inspired on the effects which have the Van Allen Belts on the trajectory of the spacecraft, it is mandatory to mention a thorough guide [17] from which the thesis has been inspired.

Nowadays, launcher vehicles do not place electric satellites in GEO straight-away. Normally, they are located in orbits closer to the Earth from which they are able to do the transfer. Generally, these intermediate orbits are ei-ther LEO or Geostationary Transfer Orbits (GTO). Therefore, these orbits are the initial point from which the low-thrust maneuver takes place. Focus-ing on the LEO to GEO transfers [18], even though their transfer times are very high, they are attainable according to [19] by following spiral strategies which allow performing the transfer without the need of any optimization tool. In [20], the matter of low-thrust spiral trajectories is studied into detail. Instead, if one looks at transfers done from GTO to GEO, the field of study is much wider, justifying the fact that these kinds of transfers are more practi-cal from the operational point of view [21,22,23]. Low-thrust maneuvers are not just utilized for transfers from LEO or GTO to GEO. The research on this topic evolves further over time and there are already some studies in which low-thrust propulsion systems are used for reaching orbits around the moon [24, 25,26,27,28]. Homologue studies exist for interplanetary orbits using low-thrust performance as well [29,30].

The existing methods to solve the wide variety of problems described above are almost endless. As explained in [6], most of them have limitations in terms of accuracy or generalization, as they do not offer a wide range of pos-sible applications. Even though covering them all is a task nearly impospos-sible to achieve, it deserves special mention the standard numerical one devel-oped in the 70s by Lester L. Sackett (and his associates) while he was at MIT Lincoln Labs: SEPSPOT. The program computes optimal planetocentric tra-jectories using the techniques of optimal control and orbital averaging [31].

1.5 Structure of the Thesis

This document is organized and structured as follows:

• CHAPTER2gives an overview of the radiation environment in which lays the groundwork of the following Chapters. In the beginning, some

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1.5. Structure of the Thesis 9

transversal general concepts are presented as well as the reference frames which are used to develop the radiation model explained in the next Chapter. Before explaining the Van Allen Radiation Belts, which is one of the main notions of this thesis, a wide detailed outline of the diverse types of radiation is described. Finally, a degradation model of the so-lar cells is given in order to measure and estimate the total equivalent fluence, which is going to be the variable to minimize under this study, in other words, the shooting function.

• CHAPTER3introduces the radiation model used throughout this the-sis in order to build the aforementioned shooting function to minimize the total equivalent fluence during its optimal trajectory. Goals of the model are presented along with the assumptions and simplifications which have been done. Listed afterward are the known issues and lim-itations of this model.

• CHAPTER4provides the mathematical basis of the thesis. It describes the dynamic model of the system and the statement of the problem, de-riving the constraints needed to solve it. In addition, a full review of optimal control theory has been included. The structure of the solver which has been used for the analysis is briefly outlined and the radia-tion optimal problem has been stated. Some validaradia-tions of the model used confronting with SPENVIS and AE9/AP9/SPM tool have been reported. Lastly, some preliminary results of the thesis are stated to finalize the Chapter.

• CHAPTER5summarizes the main findings of this research and it in-cludes a set of recommendations for future work.

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Chapter

2

The Radiation Environment

"This is Major Tom to Ground Control. . . "

- David Bowie, Space Oddity

T

HE concept of radiation in space was born less than seventy years ago once the Van Allen Belts were detected for the first time in history. Be-fore that time, the only manifestations of the presence of radiation in space were unexplained phenomena such as the aurora borealis or the deforma-tion of the ionized tail of comets caused by the solar wind [32]. Nowadays, the field of space radiation has been extensively studied and its sources com-pletely understood, even though there are still some questions that Science must find the answers to, for instance the high-energy cosmic rays1.

Life on Earth is well protected from deep space’s environmental threats by mainly three different mechanisms: the Earth’s atmosphere, the Earth’s mag-netic field and the solar winds. They will be explained in Section2.1 along with the different types of radiation encountered in deep space. This environ-ment degrades electronic systems and onboard equipenviron-ment, in particular, and creates radiobiological hazards during manned space flights [32,33]. There-fore, as new missions arise with the aim of exploring barely known deep space’s locations, such as Mars, other exoplanets (like Kepler2_{, which is a} space observatory launched by NASA in 2009 to discover Earth-size planets orbiting other stars), or Near Earth Objects3 (NEO), the satellites will face lengthened radiation exposure once they abandon the atmosphere.

This Chapter provides a brief introduction of the concepts brought in above and its main goal is to make the reader familiar with some theory of capital

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2_{https://www.nasa.gov/mission_pages/kepler/overview/index.html} _{last visited on}

27thof May, 2018

3_{Near-Earth Objects are comets and asteroids that have been nudged by the gravitational}

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12 Chapter 2. The Radiation Environment

importance regarding the radiation environment that will serve as the base for the following Chapters.

2.1 Types of Radiation

In the last paragraphs, the concept of radiation has been introduced and its main effects shortly explained. Thus, the next logical step is to describe the different sources of radiation that a spaceship would encounter once it trav-els to the outer space, far away from the natural protection of the Earth’s atmosphere. At this point, it is crucial to remark the two principal groups in which radiation is divided. The non-ionizing radiation is the one in which particles impart energy on to the atoms and molecules with which it interacts but does not strip off electrons, so it does not harm radiosensitive devices by itself. Hence, it has not been taken into account for the sources listed below. Furthermore, it is cataloged into a low energy radiation category and it is the easiest to protect aganist4 (UV radiation, for instance). On the contrary, ionizing radiation deposits energy onto the atoms and molecules with which it interacts, causing electrons to be lost. The resulting ions, or charged particles, give this form of radiation its name. They can be identified mainly as three:

• Galactic Cosmic Rays: they are created outside the solar system. They are composed of ionized atoms, independently of their number of pro-tons. Despite their associated very low flux levels, they contribute to severe ionization once passed through matter due to their fast velocity (close to the speed of light) and their heavy element composition. For the most part, the Earth’s magnetic field provides shielding for space-craft from galactic cosmic radiation. However, cosmic rays have free access over the polar regions where the magnetic field lines are open to interplanetary space5.

• Solar Cosmic Radiations: consist of two components, the low energy solar wind particles that flow constantly from the sun, and the highly energetic Solar Particle Events which are mainly high energy electrons, protons and alpha particles ejected into space. Prior to the coronal mass ejections present at the neighborhood of solar flares (Figure 2.1), the interplanetary shock waves boost these particles up almost to the speed of light.

4_{https://goo.gl/7jbXbP}_{last visited on 27}th_{of May, 2018}

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2.2. Van Allen Belts 13

• Trapped Radiation: referred normally as Van Allen Belts, it consists of energetic electrons and ions (mainly protons) which execute quasi-periodic trajectories under the constraining influence of the geomag-netic field [34]. This is the most predictable source of radiation even though it has an stochastic component which depends on the former sources. Hence, this thesis mainly focuses on this kind of radiation to implement the model explained in Chapter 3. Therefore, Section 2.2

only targets this kind of radiation and it analyses the mechanisms by which the trapped particles are governed.

FIGURE 2.1: Solar flare. Image from https://goo.gl/

hM7Fk8.

2.2 Van Allen Belts

Even though the Van Allen Radiation Belts were detected for the first time back in 1958, new missions arise nowadays with the purpose of providing new insights about it and its interactions with the Sun. One great example is the NASA’s Van Allen Probes mission [35].

The rotation of the Earth’s molten iron core results into the creation of elec-tric currents that produce magnetic field lines around the Earth from North to South like a magnet does. The aforementioned magnetic field extends sev-eral tens of thousands of kilometers into space, protecting the Earth from the charged particles of the solar wind and cosmic rays that would other-wise strip away the upper atmosphere. A solar wind is a constant stream of particles produced by the Sun, containing a wide range of elements (mainly protons and electrons), and its associated activity varies in intensity with the amount of surface activeness on the Sun. The Earth’s magnetosphere shel-ters the continuous barrage of particles and some of them get trapped in the magnetic field. They give the shape of two toroidal volumes surrounding the Earth called the Van Allen radiation belts, as depicted in Figure2.2. The inner

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belt contains a fairly stable population of protons with energies exceeding 10 MeV (up to 500 MeV). The outer belt contains mainly electrons with energies from 0.04 MeV up to 10 MeV [36].

FIGURE2.2: Simplified representation of Van Allen Belts. Im-age fromhttps://goo.gl/RKMHAj.

Of special importance for LEO is the so-called South Atlantic Anomaly (SAA), where the fringes of the inner proton radiation belt reach down to altitudes of 200 km over the southern Atlantic Ocean off the coast of Brazil, as depicted in Figure2.3. This behavior reflects the displacement of the axis of the geo-magnetic (dipole) field by about 450 km with respect to the axis of the geoid with a corresponding distortion of the magnetic field. This region accounts for up to 90% of the total exposure in LEO.

The charged particles which compose the belts circulate along the Earth’s magnetic lines of force. Their motions are a blend of three periodic motions which take place simultaneously6:

• A fast rotation (or "gyration") around magnetic field lines, typically thousands of times each second. Like the motion of planets around the Sun, this motion too can sustain itself with no energy input, and can therefore (in principle) persist for a long time. Opposite charges circle in opposite directions; around a field line pointing towards the viewer, ions circle clockwise, electrons counterclockwise.

• A slower back-and-forth bounce along the field line, typically lasting 1/10 second. As such particles circle their guiding field line, the "guid-ing center" of their rotation generally slides up or down that line, creat-ing a typical spiral pattern. However, a subtle interaction causes the spi-raling particle to be repelled from regions of the stronger magnetic field,

6_{https://www-spof.gsfc.nasa.gov/Education/wtrap1.html}_{last visited on 28}th_{of May,}

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2.3. Reference Frames 15

where field lines converge. This point where the particles are pushed back is called "mirror point".

• In addition to the rapid rotation around field lines and the back-and-forth "bounce" motion, trapped particles also undergo a slow "drift", by which they jump from one field line onto another one nearby, similar to the original one but slightly rotated around the Earth’s magnetic axis.

FIGURE2.3: South Atlantic Anomaly (SAA). Image retrieved

from SPENVIS.

The general motion of a particle trapped into the Earth’s magnetic field ex-plained above is not simple to imagine. For a better understanding of the reader, this concept is shown in the Figure2.4.

2.3 Reference Frames

In the following lines, some important reference frames used during this the-sis are described and their relations with each other derived.

2.3.1 Geocentric Equatorial Inertial Reference System

The Geocentric Equatorial Inertial Reference System (GEI) has its Z axis par-allel to the Earth’s rotation axis (positive to the North) and its X axis towards the First Point of Aries (the direction in space defined by the intersection be-tween the Earth’s equatorial plane and the plane of its orbit around the Sun (the plane of the ecliptic). This system is (to first order) fixed with respect to

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FIGURE 2.4: Motion of charged particles captured in the ge-omagnetic trap. Image taken fromhttps://www-spof.gsfc.

nasa.gov/Education/Iradbelt.html.

the distant stars. It is convenient for specifying the orbits (and hence loca-tion) of Earth-orbiting spacecraft as one can specify a Keplerian orbit in this frame. However, note that the GEI system is subject to second-order change with time owing to the various slow motions of the Earth’s rotation axis with respect to the fixed stars. Thus for GEI coordinates one must specify the date (normally termed the epoch) to which the coordinate system applies. One ex-ample is the standard astronomical epoch known as J2000.0, which is 12:00 UT on 1st January 2000. Hence, in this case, a suffix comes along with the acronym of this reference frame, i.e GEI2000.

2.3.2 Geographic Reference System

The Geographic Reference System (GRS) has its Z axis parallel to the Earth’s rotation axis (positive to the North) and its X axis towards the intersection of the Equator and the Greenwich Meridian. Thus it is convenient for specifying the location of ground stations and ground-based experiments as these are fixed quantities in the GRS system.

2.3.3 Geomagnetic Reference System

The Geomagnetic Reference System (MAG) gathers capital importance due to the fact that trapped particles follow the Earth’s magnetic field lines, as seen in the last Section. As stated in [37], the first approximation for the mag-netic field of the Earth is a dipole model. The Geomagmag-netic Reference System (MAG) has its Z axis parallel to the Earth’s magnetic dipole axis (positive North) and its Y axis is the intersection between the Earth’s equator and the geographic meridian 90 degrees east of the meridian containing the dipole axis. The X-axis completes the right-hand system.

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2.3. Reference Frames 17

2.3.4 Coordinate Transformations

Prior to explain the relations and transformation procedures between the ref-erence systems described above, a scheme of them has been depicted in Fig-ure2.5.

FIGURE2.5: Coordinate Frames. Retrieved from SPENVIS.

Once the coordinates system have been shown, the transformation proce-dures are explained. This process has been taken from [38]. The transforma-tions described in the following sectransforma-tions are presented as matrices, which are either a simple rotation matrix (a rotation of angle ζ about one of the princi-pal axes) or are the products of simple rotation matrices. These matrices have only two degrees of freedom and so only two parameters are needed to spec-ify the nine elements in the matrix. These two terms can be the rotation angle and the rotation axis: X, Y or Z, being these three the new axes expressed as unit vectors in the old coordinate system. Thus, a rotation matrix can be specified as

E= hζ, axisi (2.1)

and specify a general product of a matrices as

T=E1E2= hζ1, axis1i · hζ2, axis2i (2.2) Inversion is straightforward:

E−1= h−ζ, axisi (2.3)

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Consequently, a rotation of an angle ζ about the Z axis would be expressed in matrix form as [38] E3 = hζ, Zi =    cos ζ sin ζ 0 −sin ζ cos ζ 0 0 0 1    (2.5)

Analogously, the rotation matrices of angles β and γ about the X and Y axis, respectively read E1= hβ, Xi =    1 0 0 0 cos β sin β 0 −sin β cos β    (2.6) E2 = hγ, Yi =    cos γ 0 sin γ 0 1 0 −sin γ 0 cos γ    (2.7)

Transformations between three geocentric systems defined above it can be broken down into two fundamental transformations which are described in the following lines. The remaining transformations can then be calculated by matrix operations as shown in Table2.1.

TABLE2.1: Breakdown Coordinate Transformations.

From To GEI GRS MAG GEI I T−₁1 T−₁1T−₂1 GRS T1 I T−₂1 MAG T2T1 T2 I GEI to GRS T1 = hθ, Zi (2.8)

This matrix corresponds to a rotation in the plane of the Earth’s geographic equator from the First Point of Aries to the Greenwich meridian. The rotation angle θ is the Greenwich mean sidereal time. This can be calculated using the following formula [43] θ =100.461+36000.770T0+15.04107UT (2.9) where T0 = MJD−51544.5 36525.0 (2.10)

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2.4. Concepts of Interest 19

GRS to MAG

T2= hφ−90◦, Yi · hλ, Zi (2.11)

The two rotation are: (i) rotation in the plane of the Earth’s equator from the Greenwich meridian to the meridian containing the dipole pole, (ii) rotation in that meridian from the geographic pole to the dipole pole. The angles θ and λ are given by the following equations

λ=arctanh 1 1 g1 1 (2.12) φ=90.0−arcsing 1 1cos λ+h11sin λ g0₁ (2.13)

where the values of h1₁, g1₁and g0₁are the first order (i.e dipole) coefficients of the International Geomagnetic Reference Field (IGRF), adjusted to the time of interest (geomagnetic field changes continuously).

2.4 Concepts of Interest

In the present Section, some definitions used repeatedly during this docu-ment are presented in order to make the reader familiar with them. As stated in [37], the following concepts are referred to particles of a large range of energy E, disregarding their position in space. As explained in Section2.2, the particles trapped in the belts are constantly moving. Therefore, quanti-ties such as flux or fluence are dependent on the position in which they are measured. In the analysis reported in this thesis, for the sake of simplicity, that spatial dependency has been disregarded. If it was taken into account, then the computation of Van Allen Belts would approximate the reality with a tighter margin of error.

Definition 2. (Unidirectional Differential Intensity, J(E, θ, φ, t)) is the flux [# cm−2]7 of a given energy E per unit energy internal dE in a unit solid angle dΩ [sr] (dΩ = 2π cos θdθdφ) about the direction of observation. The unit of the Unidirectional Differential Intensity J is [# cm−2s−1sr−1keV−1] for protons and electrons, while it changes for heavy ions (check [17] for more details). Definition 3. (Omnidirectional Differential Flux, dφ_dE(E,t)) of just Differential Flux is defined as dφ(E, t) dE = Z 4πJ (E, θ, φ, t)dΩ [# cm−2s−1keV−1] (2.14) where 4π is the surface of a hypothetical sphere.

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Definition 4. (Omnidirectional Integral Flux, φ≥E(t)or φ(t)) of just Integral Flux is defined as φ(t) = Z +∞ E dφ(E, t) dE dE [# cm −2_s−1_] _(2.15) Definition 5. (Omnidirectional Differential Fluence, dΦ_dE(E)) of just Differential Fluence is defined as dΦ(E) dE = Z t_f t0 dφ(E, t) dE dt [# cm −2_keV−1_] _(2.16) Definition 6. (Omnidirectional Integral Fluence,Φ≥E orΦ) of just Integral Flu-ence is defined as Φ= Z t_f t0 Z +∞ E dφ(E, t) dE dEdt [# cm −2_] _(2.17)

2.5 Degradation Model of Solar Cells

Solar cell degradation in space is caused primarily by incident protons and electrons either trapped in the Van Allen Belts or ejected in solar events. In planning a space mission, engineers need a method of predicting the ex-pected cell degradation in the space radiation environment. This is not a simple calculation, however, because the rate of degradation for a given type of cell depends on the energies of the incident protons and electrons. In ad-dition, the front surface of the cell is usually shielded by coverglass, and the back surface by the substrate material of the cell and the supporting array structure, so that the incident particle spectrum is "slowed down" before it impinges on the active regions of the cell. Finally, different kinds of cell technologies respond differently to irradiation depending on the materials used, the thickness of the active regions, and the types and concentrations of dopants employed [39]. For this analysis, silicon solar cells have been re-ported.

A solar cell it is basically characterized by three parameters: maximum power (Pmax), open circuit voltage (Voc) and cell short circuit intensity (Isc). A generic current-voltage curve for a solar cell has been represented in Figure2.6when it comes to silicon solar cells. In this thesis, the degradation model of solar cells has been taken essentially as a tool in order to build the model exten-sively explained in the next Chapter. Therefore, only the Equivalent Fluence Theory (EFT), developed by JPL [39], is properly explained in this Section. For further details about solar cell degradation models, the reader is referred to [37,39].

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2.5. Degradation Model of Solar Cells 21

Although the definitions provided in the last Section are correct, the solar cell degradation effect has not been taken into account. Therefore, in the follow-ing lines, it is explained how those formulas vary when this phenomenon is added. This variation arises as a result of the Equivalent Fluence Theory, which is explained below. First of all it is necessary to remark that in this theory, the monoenergetic particle of 1 MeV electron is taken as a reference. The reason for this decision are few but it especially matters the fact that it is a significant component in space radiation and it can be produced conve-niently in a test environment [39]. As a result, 1 MeV electron fluence (Φ1MeV_e ) remains as the damage equivalent fluences which describe silicon solar cell degradation. For the case of protons, on the other hand, this damage is quan-tified at 10 MeV. For this reason, the proton damage ratio must be scaled in order to have an equivalent damage at a certain energy (1 MeV), a fact that gives the name to this theory. The EFT uses this idea to assess the solar cell performance during a generic space transfer.

FIGURE2.6: Current-Voltage curve. Retrieved from [37].

The Relative Damage Coefficient (RDC), De,p(E, κ), is a factor which repre-sents the damage ratio in the cell: it depends on the energy level, E, and the coverglass thickness, κ. Obviously, this coefficient is different for pro-tons/electrons and it also changes with the solar cell. Moreover, RDC values depend on the operational mode of the photovoltaic (Pmax, Vocor Isc). Hence, the evaluation of the equivalent fluences for electrons/protons yields

Φ1MeV e = Z ∞ Ee dΦe(E) dE ·De(E, κ)dE (2.18) Φ10MeV p = Z ∞ Ep dΦp(E) dE ·Dp(E, κ)dE (2.19)

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where the subindexes e and p stand for electrons and protons, respectively. Analogously, Emin_e is the minimum energy selected for the electron fluence integration while Emin

p is the one for protons. In order to have the equivalent fluences, as commented before, it is needed to have both quantities at the same energy level. Thus, a conversion parameter (αp−e) has been defined for the proton equivalent fluence, which transforms the 10 MeV proton fluence to the fluence that 1 MeV electrons would have.

Φ1MeV

p =αp−e·Φ10MeVp (2.20) The total equivalent fluence, hence, is computed as follows

Φ1MeV

total =Φ1MeVe +Φ1MeVp (2.21) This quantity is a key feature of this thesis. As it will be explained in detail later, it is the one which will be minimized while calculating the radiation optimal trajectory. In order to set down some considerations and translate the last few paragraphs into numbers, the Spectrolab XTJ solar cell has been selected regarding this work, whose conversion parameters are shown in Ta-ble2.2.

TABLE2.2: Spectrolab XTJ Conversion Parameters.

Parameter αe−p Pmax 830 Vop 782 Isc 442

Moreover, it has been considered that the photovoltaic is working at Pmax conditions for the calculations carried out during this research. Therefore,

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23

Chapter

3

Flux Distribution Model of Van

Allen Belts

"If you want to find the secrets of the universe think in terms of energy, frequency and vibration"

- Nikola Tesla, 1856 - 1943

S

HORTLYafter its discovery in 1958, the radiation belts became one of the most intriguing goals to pursue, not only in order to understand but model them in a faithful manner. The main difficulty lies in the fact that the undergoing processes which characterize them are highly stochastic. For in-stance, they are strongly dependent on solar cycles and solar activity, being this latter another stochastic process itself. Therefore, the closest approach to represent their behavior is through statistical methods. It can be done by gathering huge amount of data throughout the appropriate instruments aboard suitable satellites. This collected database is processed, analyzed, coded and, as a result, different radiation models are developed. The latest one released has been the AE9/AP9/SPM, or International Radiation Envi-ronment Near Earth (IRENE) [17], as the predecessor of the AE8/AP8/SPM. The main improvements regarding the last version (version 9) include a nar-rower spatial grid and the quantification of uncertainty due to both space weather and instrument errors [44].

3.1 Goals of the Model

The radiation model developed throughout this thesis is explained through-out this Chapter and it has straightforward goals. Firstly, it comes motivated by the need of a method to measure the radiation flux generated by elec-trons and protons encapsulated in the Van Allen belts while performing a general transfer in Earth centered environments. When the model was first

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24 Chapter 3. Flux Distribution Model of Van Allen Belts

conceived, another requisite was to develop it in an efficient manner in or-der to get fast and robust outputs. Simultaneously, it had to be simple. This comes as a high-level requirement to achieve a solution of compromise be-tween computational time and trustworthy results. The reason behind it lies in the fact that the present model will be integrated into a solver developed at Politecnico di Milano to compute radiation optimal trajectories minimiz-ing the radiation which encounters the satellite through its transfer path. The indirect approach has been applied to solve the optimization problem. This method may need large computation times, so the fact of developing the ra-diation module as simple as possible but in a consistent and robust way is a priority.

3.2 Assumptions and Simplifications

This Section provides the reader an overview of the main assumptions and simplifications done during the development of this model. In fact, these suppositions will drive most of the shortages listed in Section 4.7 because they fundamentally are unavoidable consequences.

The first and the strongest assumption which has been taken into account is the two-dimensional frame in which the model has been established. Even though the design process will be fully detailed in Section3.3, it can be said in advance that the spatial grid used was projected onto the equatorial plane, i.e. orbital inclination equal to zero. Therefore, due to the strategy used for map-ping the differential fluxes explained in the next Section, this model results to be symmetrical around latitude and longitude. Thus, for the sake of simplic-ity, and owing to the fact that the spacecraft’s instantaneous position vector provided by the solver comes in GEI coordinates as well, the present radi-ation model has been computed in the same reference system. Despite the fact that the current model could be perfectly suitable for three-dimensional frame (like the aforementioned solver), it will be only accurate as long as the Z coordinate remains close to zero.

Softer but not less important considerations have been included. One of them comprehends the epoch selected for the computation of the built-up model. An iterative analysis has been done to decide on this matter. As commented before, solar cycles affects the behavior of the Van Allen belts. Consequently, some comparisons are needed to pick a certain epoch accordingly. In Fig-ure 3.1, where the limits for proton energies Ep ∈ [0.1, 200] MeV and alti-tudes from h ∈ [200, 60000]km. The analysis has been computed through AE9/AP9/SPM tool. It has only been represented for the protons since that is the main source of radiation in terms of flux. The tights differences shown in

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3.2. Assumptions and Simplifications 25

that Figure justify some freedom margin to select the aforementioned epoch. In this thesis, it has been chosen as the 1stJanuary of 2018, 00:00:00 UT.

FIGURE 3.1: Proton averaged volume flux comparison over 11-years solar cycle.

Another consideration which has a great impact on the results is the model of solar cell which has been used (see Section2.5). Provided by SPENVIS, it maps the RDC as a piece-wise heaviside function instead of as a continuous one. RDC function for protons has been depicted in Figure 3.2 for diverse coverglass thicknesses. Its limits in energy have been shorten in the Figure to be able to see the heavyside behavior, even though this functions map the entire energy spectrum commented previously. It is straightforward to figure the approximation errors adjoined to this case. Anyhow, it has been kept in this form due to the flexibility it grants when variations of the coverglass thickness parameter are needed.

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3.3 Computational Design Strategy

Previous to explain the model’s baseline design, it is mandatory to go through some basic IRENE’s concepts. One of them is the diversity of run modes that AE9/AP9/SPM tool includes. It is of capital importance to understand what does each run mode account for and which one is the most suitable for a certain analysis. These run modes are [17]:

• Mean or Percentile: Mean mode provides a quick estimate of the envi-ronment and effects. Percentile mode is appropriate only for comparing with measurements at a given location and energy. The internal model flux map behavior remains static throughout the run, while the value initialization is either the mean or percentile. The output of this run mode is the mean or selected percentile.

• Perturbed Mean: this run mode is appropriate for cumulative/integrated quantities such as fluence, for instance. The internal model flux map be-havior remains static throughout the run, while the value initialization accounts for the mean values with random perturbations for each sce-nario. Its outputs are confidence intervals based on model uncertainties depending on the number of scenarios analyzed.

• Monte Carlo: it provides an estimate of uncertainty in time-varying quantities. The internal model flux map behavior evolves as time pro-gresses while its initialization values are the mean values with random perturbations for each scenario. Its outputs are confidence intervals in-cluding space weather based on the number of scenarios analyzed. Regarding the type of computations needed for the model under construc-tion (flux and fluences), the perturbed run mode with 100 scenarios has been used. It has been considered the 95th percentile as aggregation mode. The logic which has been followed to build the radiation model under study is summarized in Figure 3.3 and its processes are explained in the successive lines.

The first process comprehends the initialization of the main parameters of interest. These are the following:

• Vector of altitudes h. Ranging from 200 km to 60000 km with a step size of δ=39.86 km, or in a general way, N discretizations.

• Electron and proton vectors, Eeand Epwith a generic number of

com-ponents M, same for both. Their limits are Ee = 0.04−10 MeV and Ep = 0.1−200 MeV, respectively. It is important to notice that plasma has not been included in this analysis (particles below 0.04 MeV energy

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3.3. Computational Design Strategy 27

levels) due to the fact that these particles do not contribute to a remark-able solar cell degradation [48].

• The Julian days during which the ephemeris at the hkaltitude is run, Jd. In this work, 92 Julian days have been considered for all the iterations starting from the 1stof January 2018.

• Right Ascension of Ascending Node (RAAN), the argument of perigee, the true anomaly at the departure, inclination, and eccentricity are all equal to zero. Thus, all the ephemeris are contained in the equatorial plane and they are circular orbits.

The parameter k in the Figure3.3simulates the for loop in the code. Hence, in this very case, k=1, 2, ..., N, being N=1500. After the initialization of the parameters, with an altitude hkdefined already, the circular trajectory during 92 Julian days is calculated and a .txt file is generated in the suitable format for the posterior analysis through IRENE models.

FIGURE3.3: Design strategy flowchart.

In order to implement the ephemeris, the basic equations of orbital mechan-ics have been coded. One essential remark comes motivated by the sample times, which are dependent on the altitude. In accordance to [49], the recom-mended (and utilized) ones are shown in Table3.1, where RE = 6378 km is the Earth’s radius.

Once the .txt file with the ephemeris has been created, it is sent to the AE9 and AP9 modules, in which it is analyzed and the results written into .txt files. It is noteworthy to keep in mind the type of analysis done at this point. Both

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modules are run at perturbed mode with 100 scenarios while 95th _percentile is considered. Once the differential flux is calculated for both electrons and protons, the average value over the entire trajectory (92 days) is taken as the

TABLE3.1: Recommended Sample Times.

Interval Sample Time [s] hk ≤1.3RE 10

1.3RE <hk ≤2RE 60 2RE <hk ≤4RE 300

hk >4RE 900

final output. These values of the differential fluxes are computed for every energy level predefined by Ep and Ee, respectively. Therefore, they are vec-tors with the same number of components that the aforementioned energy vectors (M components). These vectors are the responsible ones of filling the differential flux surfaces saved in the workspace gradually as matrices: Fpfor

protons and Fefor electrons. In a mathematical way, this yields

Fp(1 : M, k) = dφp(Ep, hk) dEp (3.1) Fe(1 : M, k) = dφe (Ee, hk) dEe (3.2) In the Figures3.4, the result of the detailed process described above is given. As the reader could notice, these graphs are in linear scale and they are quite steep for low energy levels, a fact that could be a major difficulty to interpo-late them.

FIGURE 3.4: Linear differential flux distribution for protons (left) and electrons (right).

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3.3. Computational Design Strategy 29

FIGURE3.5: Logarithmic differential flux distribution for pro-tons (left) and electrons (right).

For facing this problem, they have been finally represented in logarithmic scale, as shown in the Figures3.5. Then, they have been approximated through a linear interpolation and saved afterward into two structure variables, one for each particle type. To finalize the current Section, both 2-D integral flux models for protons and electrons have been depicted in Figure3.6. As can be seen in the Figures and according to [17], the inner belt extends from approx-imately hundreds of kilometers to 6000 km in altitude and is populated by high energy protons and medium energy electrons, while the outer belt, up to 60,000 km in altitude, is predominately populated by electrons.

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FIGURE 3.6: 2-D Model of Van Allen Belts for protons (top) and electrons (bottom) in GEI reference frame.

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31

Chapter

4

Low Thrust Radiation Optimal

Trajectory Problem

"Human knowledge is personal and responsible, an unending ad-venture at the edge of uncertainty"

- Jacob Bronowski, 1908 - 1974

I

NCIDENT radiation contribution of Van Allen Belts generates significant

failures within the satellite, especially in solar cells through degradation processes. Regarding this fact, reducing spacecraft’s radiation exposure is a major challenge. In this Chapter, the low-thrust transfer to GEO with fix TOF has been investigated. The main aim in this approach consists in minimizing the total equivalent fluence while the spacecraft is flying across the Van Allen Belts. The problem has been framed and analyzed using the indirect method.

4.1 Dynamic Model

Prior to formulating optimal trajectory problem, it is mandatory to set the dy-namics equations which model the spacecraft. These equations are affected by the Earth gravitational acceleration and the thrust provided by the space-craft itself, which is considered as a point mass body. As known from the literature, Kepler’s problem is a particular case derived from the two-body problem in which they interact by means of a central force. The spacecraft’s dynamics in an inertial reference frame is formulated in Cartesian coordi-nates as follows d2r dt2 = −µ r |r|3+ T m (4.1)

where r ∈ R3 _{represents the position vector, T} _∈ _R3 _{stands for the thrust} vector, m is the spacecraft’s mass and µ = G·ME is the Earth gravitational

A radiation model for power degradation estimation in earth centered satellites

P

OLITECNICO DI

M

ILANO

M

ASTER

T

HESIS

A Radiation Model for Power

Degradation Estimation in Earth

Centered Satellites

S

I

I

E

D

S

A

T

Abstract

E

Sommario

I

Acknowledgements

Contents

List of Figures

List of Tables

Notation

Acronyms

Dedicated to my parents and sister, Alberto, Lola and

Julia, whom I love tenderly.

Chapter

1

Introduction

O

1.1

Low-Thrust Optimal Trajectories

1.2

Motivation of the Thesis

1.3

Goals of the Thesis

1.4

State of the Art

1.5

Structure of the Thesis

Chapter

2

The Radiation Environment

T

2.1

Types of Radiation

2.2

Van Allen Belts

2.3

Reference Frames

2.4

Concepts of Interest

2.5

Degradation Model of Solar Cells

Chapter

3

Flux Distribution Model of Van

Allen Belts

S

3.1

Goals of the Model

3.2

Assumptions and Simplifications

3.3

Computational Design Strategy

Chapter

4

Low Thrust Radiation Optimal

Trajectory Problem

I

4.1

Dynamic Model